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A185055
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Number of representations of 5^(2n) as a sum a^2 + b^2 + c^2 with 0 < a <= b <= c.
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1
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0, 0, 2, 14, 76, 388, 1950, 9762, 48824, 244136, 1220698, 6103510, 30517572, 152587884, 762939446, 3814697258, 19073486320, 95367431632, 476837158194, 2384185791006, 11920928955068, 59604644775380, 298023223876942, 1490116119384754, 7450580596923816, 37252902984619128
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OFFSET
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0,3
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COMMENTS
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Corresponding formulas for several first primes:
p=3, a(n)=(3*3^n+2*n+1)/4 (A047926)
p=7, a(n)=(7^n-1)/6
p=11, a(n)=(3*11^n+10*n-3)/20
p=13, a(n)=(13^n-4*n-1)/8
p=17, a(n)=(17^n-1)/8
p=19, a(n)=(5*19^n+18n-5)/36
p=23, a(n)=3(23^n-1)/22
p=29, a(n)=(29^n-4n-1)/8
p=31, a(n)=2(31^n-1)/15
p=37, a(n)=(37^n-4*n-1)/8
p=41, a(n)=(41^n-1)/8
p=43, a(n)=(11*43^n+42n-11)/84
p=47, a(n)=(3(47^n-1)/23.
General formulas for a(n) depend on p mod 8 as follows:
p= 1 mod 8 , a(n)=(p^n-1)/8
p = 3 mod 8, a(n)=((p + 1)*p^n + 4*(p - 1)*n - (p + 1))/(8*(p - 1))
p = 5 mod 8, a(n)=(p^n-4*n-1)/8
p = 7 mod 8, a(n)=((p + 1)*(p^n - 1))/(8*(p - 1)).
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LINKS
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FORMULA
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a(n) = (5^n-4n-1)/8.
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EXAMPLE
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a(2)=2 because 25^2 = 9^2+12^2+20^2 = 12^2+15^2+16^2.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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